Tight estimates for eigenvalues of regular graphs

Tight Estimates for Eigenvalues of Regular Graphs

Spectra of Regular Graphs and Hypergraphs and Orthogonal Polynomials

In this paper we study the distribution of eigenvalues of regular graphs , regular hypergraphs , and biregular bipartite graphs of given girth by considering the polynomials orthogonal with respect to the measures attached to the spectra of such graphs and to the continuous spectra of their ‘universal covers’ . Our estimates are tight for Biggs graphs and generalized polygons . We also give an ...

D-Spectrum and D-Energy of Complements of Iterated Line Graphs of Regular Graphs

The D-eigenvalues {µ1,…,µp} of a graph G are the eigenvalues of its distance matrix D and form its D-spectrum. The D-energy, ED(G) of G is given by ED (G) =∑i=1p |µi|. Two non cospectral graphs with respect to D are said to be D-equi energetic if they have the same D-energy. In this paper we show that if G is an r-regular graph on p vertices with 2r ≤ p - 1, then the complements of iterated lin...

Distance-regular graphs with complete multipartite μ-graphs and AT4 family

Let be an antipodal distance-regular graph of diameter 4, with eigenvalues θ0 > θ1 > θ2 > θ3 > θ4. Then its Krein parameter q4 11 vanishes precisely when is tight in the sense of Jurišić, Koolen and Terwilliger, and furthermore, precisely when is locally strongly regular with nontrivial eigenvalues p := θ2 and −q := θ3. When this is the case, the intersection parameters of can be parametrized b...

Tight Distance-regular Graphs with Small Diameter

We prove the following bound for a k regular graph on n vertices with nontrivial eigenvalues from the interval r s n k rs k r k s Equality holds if and only if the graph is strongly regular with eigenvalues in fk s rg Nonbipartite distance regular graphs with diameter d and eigenvalues k d whose local graphs satisfy the above bound with equality for s b and r b d are called tight graphs and are...